Reflectionless measures are interesting objects because
they arise as limiting measures of spectral measures of arbitrary
Schr"odinger operators with some absolutely continuous spectrum.
In this talk, I'd like to review the definition and some background
material and then discuss more recent work, joint with Alexei Poltoratski,
on reflectionless measures.

In this talk, we consider the evolution of a tight binding wave packet propagating in a time dependent potential. We assume the potential evolves according to a stationary Markov process and show that the square amplitude of the wave packet converges to a solution of a heat equation. This is joint work with Jeff Schenker.

We discuss joint work with Jon Chaika and Helge Krueger. The main result concerns explicit criteria for the absence of absolutely continuous spectrum for Schrodinger operators whose potentials are generated by an interval exchange transformation. In particular, we provide the first example of an invertible ergodic transformation of a compact metric space for which the associated Schrodinger operators have purely singular spectrum for every non-constant continuous sampling function.